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My motivation is making general proof , instead of trying to prove special cases.

To which branch of mathematics does my question belong?

I am highly interested in irrational numbers.

Is it good idea , proving that the number in my question has algebraic degree higher than one(and why is degree greater than one implies number is irrational)?

I searched and I didn't find good references to my wondering.

Mahmoud albahar
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    This theme recurs. The argument from my answer to a slightly different question works as is here as well. That method is arguably unnecessarily high-browed in that Galois theory is not strictly necessary. In this thread we still use field theory, and that may be necessary to settle questions like this (when the number of square roots grows, and purely arithmetical techniques become unwieldy). – Jyrki Lahtonen Dec 08 '22 at 15:33
  • @JyrkiLahtonen Thanks – Mahmoud albahar Dec 08 '22 at 15:35
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    This is close to being an exact match. Check out the links in Qiaochu Yuan's answer. Olympiad style answers are promised. I haven't checked exactly what that means, but it usually implies that less advanced theory is required (but technical prowess goes up to compensate). – Jyrki Lahtonen Dec 08 '22 at 15:39

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Call $p_1:=2$, $p_2:=3$ and so on, $p_n$ is the $n$th prime number. Consider the set

$$E_n:=\{\varepsilon_1 \sqrt{p_1}+\ldots+\varepsilon_n \sqrt{p_n}:\ \varepsilon_i∈\{-1,1\}\}$$

so $E_n$ contains all $2^n$ possible combinations of sums and differences of the square roots $\sqrt2$, $\sqrt3$, $\ldots$, $\sqrt{p_n}$.

Now, consider the polynomial $$p_n(x):=\prod_{\alpha\in E_n} (x-\alpha)$$

This is called the $n$th Swinnerton-Dyer polynomial, it is an irreducible polynomial with coefficient in $\mathbb Z$. So all its roots are irrational numbers with degree $2^n$

Robert Z
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alvoi
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    This is a fine method, of course (I upvoted). Proving that the polynomial is irreducible may not be too easy :-) I didn't know these were called Swinnerton-Dyer polynomials! – Jyrki Lahtonen Dec 08 '22 at 15:35
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    Thanks and I wanna ask you, how do we prove that coefficients are in $\mathbb{Z}$ And how to prove irreducibility? I want you to guide me to good references. – Mahmoud albahar Dec 08 '22 at 15:37
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    Yeah, if I remember correctly there is a bit of Galois theory involved! – alvoi Dec 08 '22 at 15:37